New Results for Generalized Gronwall Inequalities and Their Applications

Abstract

In succession to our earlier work, we further provide some new generalized Gronwall inequalities and apply these inequalities to the study of qualitative estimations of solutions to certain fractional differential equations.

1. Introduction

It is well known that the Gronwall inequality contributes significantly to research on many differential and integral equations. An increasing number of generalizations of this inequality have been made in recent years to derive qualitative properties of solutions to various fractional differential equations. One remarkable result was obtained by Ye et al. in 2007 and is presented below.

Theorem 1 (see [1, Theorem 1]). For any 𝑡∈[0,𝑇), 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀(𝑡−𝑠)^𝛽−1𝑢(𝑠)𝑑𝑠, (1) where all functions are nonnegative and continuous. The constant 𝛽>0. 𝑏 is a bounded and monotonically increasing function of [0,𝑇); then 𝑢(𝑡)≤𝑎(𝑡)+^𝑡₀^∞_𝑛=1𝑏(𝑡)Γ𝛽^𝑛Γ()𝑛𝛽𝑡−𝑠^𝑛𝛽−1𝑎(𝑠[).)𝑑𝑠,𝑡∈0,𝑇 (2)

This result continuously extends and improves and has been widely used in many studies (e.g., see [1–8]). Among them, our previous research [5] provides the following generalizations to deal with complex fractional differential equations.

Theorem 2 (see [5, Theorem 1.4]). For any 𝑡∈[0,𝑇), 𝑢(𝑡)≤𝑎(𝑡)+^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑢(𝑠)𝑑𝑠, (3) where all functions are nonnegative and continuous. The constants 𝛽_𝑖>0. 𝑏_𝑖(𝑖=1,2,…,𝑛) are bounded and monotonically increasing functions of [0,𝑇). Then 𝑢(𝑡)(𝑡)+≤𝑎_{∞𝑘=1𝑛1′,2′,…,𝑘′=1}∏^𝑘_𝑖=1𝑏_𝑖′(𝑡)Γ𝛽_𝑖′Γ∑^𝑘_𝑖=1𝛽_𝑖′×^𝑡₀(𝑡−𝑠)^{∑𝑘𝑖=1𝛽𝑖′−1}𝑎(𝑠),[).𝑑𝑠𝑡∈0,𝑇 (4)

Theorem 3 (see [5, Theorem 1.5]). For any 𝑡∈[1,𝑇), 𝑢(𝑡)≤𝑎(𝑡)+^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡₁𝑡ln𝑠^𝛽𝑖−1𝑢(𝑠)𝑠𝑑𝑠, (5) where all functions are nonnegative and continuous. The constants 𝛽_𝑖>0. 𝑏_𝑖(𝑖=1,2,…,𝑛) are bounded and monotonically increasing functions of [1,𝑇). Then 𝑢(𝑡)(𝑡)+≤𝑎_{∞𝑘=1𝑛1′,2′,…,𝑘′=1}∏^𝑘_𝑖=1𝑏_𝑖′(𝑡)Γ𝛽_𝑖′Γ∑^𝑘_𝑖=1𝛽_𝑖′×^𝑡₁𝑡ln𝑠^{∑𝑘𝑖=1𝛽𝑖′−1}𝑎(𝑠)𝑑𝑠𝑠,𝑡∈[1,𝑇). (6)

In this paper, we aim to discuss further issues by using the aforementioned conclusions and suitable analytical techniques according to the above facts. We will establish several new classes of generalized Gronwall inequalities in the next section. In the last section, we will qualitatively analyze certain fractional differential equations to highlight the applications of the inequalities.

2. Main Results and Proofs

We introduce the useful Young’s inequality with 𝜖>0 (see [9, page 622]); that is, for any 𝑎,𝑏>0 and 1<𝑝, 𝑞<+∞, 1/𝑝+1/𝑞=1, 𝑎𝑏≤𝜖𝑎^𝑝+𝐶(𝜖)𝑏^𝑞, (7) where 𝐶(𝜖)=(𝜖𝑝)^−𝑞/𝑝𝑞⁻¹.

The first main result and its proof procedure are presented as follows.

Theorem 4. For any 𝑡∈[0,𝑇), 𝑢(𝑡)≤𝑎(𝑡)+^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)𝑑𝑠, (8) where all functions are nonnegative and continuous. For any 𝑖∈{1,2,…,𝑛}, the constants 𝛽_𝑖>0 and 0<𝜆_𝑖<1, and the function 𝑏_𝑖(𝑡) is bounded and monotonically increasing on [0,𝑇). Then 𝑢(𝑡)≤𝑎(𝑡)+^∞_𝑘=1^𝑛_{1′,2′,…,𝑘′=1}∏^𝑘_𝑖=1𝑏̃_𝑖′(𝑡)Γ𝛽_𝑖′Γ∑^𝑘_𝑖=1𝛽_𝑖′×^𝑡₀(𝑡−𝑠)^{∑𝑘𝑖=1𝛽𝑖′−1}𝑎(𝑠),[),𝑑𝑠𝑡∈0,𝑇 (9) where 𝑎(𝑡)=𝑎(𝑡)+^𝑛_𝑖=11−𝜆_𝑖𝜆_𝑖𝜀^{𝜆𝑖/(1−𝜆𝑖)}𝑏_𝑖(𝑡)×^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)^{1/(1−𝜆𝑖)}𝑏̃𝑑𝑠;_𝑖(𝑡)=𝜀⋅𝑏_𝑖(𝑡). (10) Here 𝜀 is an arbitrary given positive number.

Proof. By Young’s inequality, 𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)≤𝜀⋅𝑢^𝜆𝑖(𝑠)^1/𝜆𝑖+𝜆_𝑖𝜀^{𝜆𝑖/(1−𝜆𝑖)}⋅1−𝜆_𝑖⋅𝑐_𝑖(𝑠)^{1/(1−𝜆𝑖)}(𝑠)+𝜆=𝜀⋅𝑢^𝑖𝜀^{𝜆𝑖/(1−𝜆𝑖)}⋅1−𝜆_𝑖⋅𝑐_𝑖(𝑠)^{1/(1−𝜆𝑖)}, (11) which implies that, for any 𝑡∈[0,𝑇), 𝑢(𝑡)≤𝑎(𝑡)+^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)(𝑡)+𝑑𝑠≤𝑎^𝑛_𝑖=1𝑏_𝑖(𝑡)×^𝑡₀(𝑡−𝑠)^𝛽𝑖−1(𝑠)+𝜆𝜀⋅𝑢_𝑖𝜀^{𝜆𝑖/(1−𝜆𝑖)}⋅1−𝜆_𝑖⋅𝑐_𝑖(𝑠)^{1/(1−𝜆𝑖)}=𝑎(𝑡)+𝑑𝑠_𝑛𝑖=11−𝜆_𝑖𝜆_𝑖𝜀^{𝜆𝑖/(1−𝜆𝑖)}𝑏_𝑖(𝑡)×^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)^{1/(1−𝜆𝑖)}+𝑑𝑠_𝑛𝑖=1𝜀⋅𝑏_𝑖(𝑡)^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑢(𝑠)=𝑎(𝑡)+𝑑𝑠_𝑛𝑖=1𝑏̃_𝑖(𝑡)^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑢(𝑠)𝑑𝑠. (12) We immediately derived estimation (9) by using Theorem 2. This estimation completes the proof of Theorem 4.

Our second result can be proved in the same manner by applying Young’s inequality and Theorem 3. The proof procedure is similar to that of Theorem 4 and is omitted here.

Theorem 5. For any 𝑡∈[1,𝑇), 𝑢(𝑡)≤𝑎(𝑡)+^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡₁𝑡ln𝑠^𝛽𝑖−1𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)𝑠𝑑𝑠, (13) where all functions are nonnegative and continuous. For any 𝑖∈{1,2,…,𝑛}, the constants 𝛽_𝑖>0 and 0<𝜆_𝑖<1, and the function 𝑏_𝑖(𝑡) is bounded and monotonically increasing on [1,𝑇). Then 𝑢(𝑡)≤𝑎(𝑡)+^∞_𝑘=1^𝑛_{1′,2′,…,𝑘′=1}∏^𝑘_𝑖=1𝑏_𝑖′(𝑡)Γ𝛽_𝑖′Γ∑^𝑘_𝑖=1𝛽_𝑖′×^𝑡₁𝑡ln𝑠^{∑𝑘𝑖=1𝛽𝑖′−1}𝑎(𝑠)𝑠,[),𝑑𝑠𝑡∈1,𝑇 (14) where 𝑎(𝑡)=𝑎(𝑡)+^𝑛_𝑖=11−𝜆_𝑖𝜆_𝑖𝜀^{𝜆𝑖/(1−𝜆𝑖)}𝑏_𝑖(𝑡)×^𝑡₁𝑡ln𝑠^𝛽𝑖−1𝑐_𝑖(𝑠)^{1/(1−𝜆𝑖)}𝑑𝑠𝑠;𝑏_𝑖(𝑡)=𝜀⋅𝑏_𝑖(𝑡). (15) Here 𝜀 is an arbitrary given positive number.

Evidently, the following two corollaries are directly obtained by using Theorems 4 and 5 when choosing 𝛽_𝑖=1 for any 𝑖∈{1,2,…,𝑛}.

Corollary 6. For any 𝑡∈[0,𝑇), 𝑢(𝑡)≤𝑎(𝑡)+^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡₀𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)𝑑𝑠, (16) where all functions are nonnegative and continuous. For any 𝑖∈{1,2,…,𝑛}, the constant 0<𝜆_𝑖<1, and the function 𝑏_𝑖(𝑡) is bounded and monotonically increasing on [0,𝑇). Then 𝑢(𝑡)≤𝑎(𝑡)+^∞_𝑘=1^𝑛_{1′,2′,…,𝑘′=1}∏^𝑘_𝑖=1𝑏̃_𝑖′(𝑡)×𝑘!_𝑡0(𝑡−𝑠)^𝑘−1𝑎(𝑠),[),𝑑𝑠𝑡∈0,𝑇 (17) where 𝑎(𝑡)=𝑎(𝑡)+^𝑛_𝑖=11−𝜆_𝑖𝜆_𝑖𝜀^{𝜆𝑖/(1−𝜆𝑖)}𝑏_𝑖(𝑡)×^𝑡₀𝑐_𝑖(𝑠)^{1/(1−𝜆𝑖)}𝑏̃𝑑𝑠;_𝑖(𝑡)=𝜀⋅𝑏_𝑖(𝑡). (18) Here 𝜀 is an arbitrary given positive number.

Corollary 7. For any 𝑡∈[1,𝑇), 𝑢(𝑡)≤𝑎(𝑡)+^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡₁𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)𝑠𝑑𝑠, (19) where all functions are nonnegative and continuous. For any 𝑖∈{1,2,…,𝑛}, the constant 0<𝜆_𝑖<1, and the function 𝑏_𝑖(𝑡) is bounded and monotonically increasing on [1,𝑇). Then 𝑢(𝑡)≤𝑎(𝑡)+^∞_𝑘=1^𝑛_{1′,2′,…,𝑘′=1}∏^𝑘_𝑖=1𝑏_𝑖′(𝑡)×𝑘!_𝑡1𝑡ln𝑠^𝑘−1𝑎(𝑠)𝑑𝑠𝑠,𝑡∈[1,𝑇), (20) where 𝑎(𝑡)=𝑎(𝑡)+^𝑛_𝑖=11−𝜆_𝑖𝜆_𝑖𝜀^{𝜆𝑖/(1−𝜆𝑖)}𝑏_𝑖(𝑡)×^𝑡₁𝑐_𝑖(𝑠)^{1/(1−𝜆𝑖)}𝑑𝑠𝑠;𝑏_𝑖(𝑡)=𝜀⋅𝑏_𝑖(𝑡). (21) Here 𝜀 is an arbitrary given positive number.

Our next task is estimating the nonnegative and continuous function 𝑢(𝑡), which satisfies, for any 𝑡∈[0,𝑇), 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑐(𝑠)𝑢(𝑠)+𝑑𝑠_𝑛𝑖=1𝑏_𝑖(𝑡)^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)𝑑𝑠, (22) where all the functions are nonnegative and continuous. For any 𝑖∈{1,2,…,𝑛}, the constants 𝛽_𝑖>0 and 0<𝜆_𝑖<1, and the functions 𝑏(𝑡) and 𝑏_𝑖(𝑡) are bounded and monotonically increasing on [0,𝑇). Put 𝑀_𝑖(𝑡)=max_{0≤𝑠≤𝑡}𝑏_𝑖(𝑠)𝑐(𝑠). (23) Suppose that 𝐴(𝑡)=𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝑐(𝑠)𝑒^{∫𝑡𝑠𝑐(𝑤)𝑏(𝑤)𝑑𝑤}(𝑠)𝐵⋅𝑎𝑑𝑠;_𝑖(𝑡)=𝑏_𝑖(𝑡)+𝑡⋅𝑏(𝑡)⋅𝑀_𝑖(𝑡)𝛽_𝑖𝑒^{∫𝑡0𝑐(𝑤)𝑏(𝑤)𝑑𝑤},𝑖=1,2,…,𝑛. (24) Obviously, 𝐵_𝑖(𝑡)≥0(𝑖=1,2,…,𝑛) are bounded and monotonically increasing functions. Moreover, we assumed that 𝐴(𝑡)=𝐴(𝑡)+^𝑛_𝑖=11−𝜆_𝑖𝜆_𝑖𝜀^{𝜆𝑖/(1−𝜆𝑖)}𝐵_𝑖(𝑡)×^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)^{1/(1−𝜆𝑖)}𝐵𝑑𝑠;_𝑖(𝑡)=𝜀𝐵_𝑖(𝑡),𝑖=1,2,…,𝑛. (25) Here 𝜀 is an arbitrary given positive number. The third main result is given as follows.

Theorem 8. For any 𝑡∈[0,𝑇), the nonnegative and continuous function 𝑢(𝑡) satisfies the inequality (22); then 𝑢(𝑡)≤𝐴(𝑡)+^∞_𝑘=1^𝑛_{1′,2′,…,𝑘′=1}∏^𝑘_𝑖=1𝐵_𝑖′(𝑡)Γ𝛽_𝑖′Γ∑^𝑘_𝑖=1𝛽_𝑖′×^𝑡₀(𝑡−𝑠)^{∑𝑘𝑖=1𝛽𝑖′−1}𝐴(𝑠),[),𝑑𝑠𝑡∈0,𝑇 (26) where the expressions 𝐴(𝑡) and 𝐵_𝑖(𝑡) are described in (25).

Proof. Suppose that 𝐼(𝑡)=𝑎(𝑡)+^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)𝑑𝑠. (27) Then, (22) transforms into the following form: 𝑢(𝑡)≤𝐼(𝑡)+𝑏(𝑡)^𝑡₀𝑐(𝑠)𝑢(𝑠)𝑑𝑠. (28) Therefore, 𝑐(𝑡)𝑢(𝑡)≤𝑐(𝑡)𝐼(𝑡)+𝑐(𝑡)𝑏(𝑡)^𝑡₀𝑐(𝑠)𝑢(𝑠)𝑑𝑠. (29) Letting 𝑢₁(𝑡)=𝑐(𝑡)𝑢(𝑡), 𝐼₁(𝑡)=𝑐(𝑡)𝐼(𝑡), and 𝐽₁(𝑡)=𝑐(𝑡)𝑏(𝑡) obtains 𝑢₁(𝑡)≤𝐼₁(𝑡)+𝐽₁(𝑡)^𝑡₀𝑢₁(𝑠)𝑑𝑠. (30) By using the classical Gronwall inequality (see [10, page 15]), we have 𝑢₁(𝑡)≤𝐼₁(𝑡)+𝐽₁(𝑡)^𝑡₀𝐼₁(𝑠)𝑒^{∫𝑡𝑠𝐽1(𝑤)𝑑𝑤}𝑑𝑠; (31) that is, 𝑢(𝑡)≤𝐼(𝑡)+𝑏(𝑡)^𝑡₀𝑐(𝑠)𝐼(𝑠)𝑒^{∫𝑡𝑠𝑐(𝑤)𝑏(𝑤)𝑑𝑤}=𝑎(𝑡)(𝑡)𝑑𝑠+𝑏^𝑡₀𝑐(𝑠)𝑎(𝑠)𝑒^{∫𝑡𝑠𝑐(𝑤)𝑏(𝑤)𝑑𝑤}+𝑑𝑠_𝑛𝑖=1𝑏_𝑖(𝑡)^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)(𝑡)𝑑𝑠+𝑏^𝑛_𝑖=1^𝑡₀^𝑠₀𝑐(𝑠)𝑒^{∫𝑡𝑠𝑐(𝑤)𝑏(𝑤)𝑑𝑤}×𝑏_𝑖(𝑠)(𝑠−𝑣)^𝛽𝑖−1𝑐_𝑖(𝑣)𝑢^𝜆𝑖(𝑣)𝑑𝑣𝑑𝑠, (32) given that ^𝑛_𝑖=1^𝑡₀^𝑠₀𝑏_𝑖(𝑠)𝑐(𝑠)𝑒^{∫𝑡𝑠𝑐(𝑤)𝑏(𝑤)𝑑𝑤}×(𝑠−𝑣)^𝛽𝑖−1𝑐_𝑖(𝑣)𝑢^𝜆𝑖(𝑣)=𝑑𝑣𝑑𝑠_{𝑛𝑖=1𝑡0𝑡𝑣}𝑏_𝑖(𝑠)𝑐(𝑠)𝑒^{∫𝑡𝑠𝑐(𝑤)𝑏(𝑤)𝑑𝑤}×(𝑠−𝑣)^𝛽𝑖−1𝑐_𝑖(𝑣)𝑢^𝜆𝑖(𝑣)𝑑𝑠𝑑𝑣≤𝑒^{∫𝑡0𝑐(𝑤)𝑏(𝑤)𝑑𝑤}×^𝑛_𝑖=1𝑀_𝑖(𝑡)^𝑡₀^𝑡_𝑣(𝑠−𝑣)^𝛽𝑖−1𝑐_𝑖(𝑣)𝑢^𝜆𝑖(𝑣)𝑑𝑠𝑑𝑣=𝑒^{∫𝑡0𝑛𝑐(𝑤)𝑏(𝑤)𝑑𝑤}_𝑖=11𝛽_𝑖𝑀_𝑖(𝑡)×^𝑡₀(𝑡−𝑣)^𝛽𝑖−1(𝑡−𝑣)𝑐_𝑖(𝑣)𝑢^𝜆𝑖(𝑣)𝑑𝑣≤𝑒^{∫𝑡0𝑛𝑐(𝑤)𝑏(𝑤)𝑑𝑤}_𝑖=11𝛽_𝑖𝑡𝑀_𝑖(𝑡)×^𝑡₀(𝑡−𝑣)^𝛽𝑖−1𝑐_𝑖(𝑣)𝑢^𝜆𝑖(𝑣)𝑑𝑣, (33) where the function 𝑀_𝑖(𝑡) is defined in (23). Combining (24), (32), and (33) yields the following inequality: 𝑢(𝑡)≤𝐴(𝑡)+^𝑛_𝑖=1𝐵_𝑖(𝑡)^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)𝑑𝑠. (34) The estimation (26) is obtained according to Theorem 4. This process completes the proof of Theorem 8.

In the same manner, the final result in this section can be obtained by applying the conclusion of Theorem 5.

Theorem 9. For any 𝑡∈[1,𝑇), 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₁𝑐(𝑠)𝑢(𝑠)𝑑𝑠𝑠+^𝑛_𝑖=1𝑏_𝑖(𝑡)^𝑡₁𝑡ln𝑠^𝛽𝑖−1𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)𝑑𝑠𝑠, (35) where all functions are nonnegative and continuous. For any 𝑖∈{1,2,…,𝑛}, the constants 𝛽_𝑖>0 and 0<𝜆_𝑖<1, and the functions 𝑏(𝑡) and 𝑏_𝑖(𝑡) are bounded and monotonically increasing on [1,𝑇). Then 𝑢(𝑡)≤𝐴(𝑡)+^∞_𝑘=1^𝑛_{1′,2′,…,𝑘′=1}∏^𝑘_𝑖=1𝐵_𝑖′(𝑡)Γ𝛽_𝑖′Γ∑^𝑘_𝑖=1𝛽_𝑖′×^𝑡₁𝑡ln𝑠^{∑𝑘𝑖=1𝛽𝑖′−1}𝐴(𝑠)𝑠,[),𝑑𝑠𝑡∈1,𝑇 (36) where 𝐴(𝑡)=𝐴^′(𝑡)+^𝑛_𝑖=11−𝜆_𝑖𝜆_𝑖𝜀^{𝜆𝑖/(1−𝜆𝑖)}𝐵^′_𝑖(𝑡)×^𝑡₁𝑡ln𝑠^𝛽𝑖−1𝑐_𝑖(𝑠)^{1/(1−𝜆𝑖)}𝑑𝑠𝑠;𝐵_𝑖(𝑡)=𝜀⋅𝐵^′_𝑖(𝑡). (37) Here 𝐴^′(𝑡)=𝑎(𝑡)+𝑏(𝑡)^𝑡₁𝑐(𝑠)𝑒^{∫𝑡𝑠𝑐(𝑤)𝑏(𝑤)(𝑑𝑤/𝑤)}⋅𝑎(𝑠)𝑑𝑠𝑠;𝐵^′_𝑖(𝑡)=𝑏_𝑖(𝑡)+ln𝑡⋅𝑏(𝑡)⋅𝑀^′_𝑖(𝑡)𝛽_𝑖𝑒^{∫𝑡1𝑐(𝑤)𝑏(𝑤)(𝑑𝑤/𝑤)},𝑖=1,2,…,𝑛. (38) Also 𝑀^′_𝑖(𝑡)=max_{1≤𝑠≤𝑡}{𝑏_𝑖(𝑠)𝑐(𝑠)}, and 𝜀 is an arbitrary given positive number.

The proof procedure of Theorem 9 is relatively similar to that of Theorem 8. Hence, the procedure will not be presented in this paper.

3. Applications

In this section, we apply the main results in Section 2 to provide qualitative conclusions for solutions of certain fractional differential equations. First, the definitions and some properties of the Riemann-Liouville fractional derivative and integral need to be recalled.

Definition 10 (see [11–16]). For any 0<𝛽<1 and a continuous function 𝑤, the 𝛽th Riemann-Liouville type fractional order derivative 𝐷^𝛽_𝑅𝑤 and the corresponding fractional integral operator 𝐼^𝛽_𝑅𝑤 are defined by 𝐼^𝛽_𝑅𝑤(𝑡)=1Γ𝛽^𝑡₀(𝑡−𝑠)^𝛽−1𝑤(𝑠)𝐷𝑑𝑠;_𝛽𝑅𝑤(𝑡)=1Γ𝑑1−𝛽𝑑𝑡^𝑡₀(𝑡−𝑠)^−𝛽𝑤(𝑠)𝑑𝑠. (39)

Lemma 11 (see [13, Lemma 2.2]). For any 0<𝛽<1 and a continuous function 𝑤, 𝐼^𝛽_𝑅𝐷^𝛽_𝑅𝑤(𝑡)=𝑤(𝑡)+𝑘⋅𝑡^𝛽−1, (40) where 𝑘 is a certain constant in ℝ.

Lemma 12 (see [17, Page 14]). For any 𝛼,𝛽>0 and a continuous function 𝑤, 𝐼^𝛼_𝑅𝐼^𝛽_𝑅𝑤(𝑡)=𝐼_𝑅^𝛼+𝛽𝑤(𝑡);𝐼^𝛼_𝑅𝑡^𝛽=Γ𝛽+1Γ𝑡𝛼+𝛽+1^𝛼+𝛽. (41)

Given the aforementioned preliminary knowledge, we consider the following initial value problem: ^𝑛_𝑖=1𝐷^{𝛽𝑛+1−𝛽𝑖}_𝑅𝑐_𝑖(𝑡)𝑢^𝜆𝑖(𝑡)+𝐷^𝛽𝑛+1_𝑅𝑢^𝜆0(𝑡)=𝑓(𝑡,𝑢(𝑡))_𝑛𝑖=1;𝐼^{1+𝛽𝑖−𝛽𝑛+1}_𝑅𝑐_𝑖(𝑡)𝑢^𝜆𝑖(𝑡)+𝐼^{1−𝛽𝑛+1}_𝑅𝑢^𝜆0(𝑡)_𝑡=0=𝛿, (42) where all functions are continuous. Moreover 𝑐_𝑖(𝑡)>0 and the constants 𝜆_𝑖,𝛽_𝑖>0 (𝑖=1,2,…,𝑛). Consider 𝜆₀>0, 𝛿∈ℝ, and 1>𝛽_𝑛+1>max_{1≤𝑖≤𝑛}{𝛽_𝑖}.

Theorem 13. Suppose that, for any 𝑡∈[0,𝑇] and 𝑦,𝑧∈ℝ, 𝑓𝑡,𝑦−𝑓(𝑡,𝑧)≤𝑐_𝑛+1(𝑡)𝑦^𝜆𝑛+1−𝑧^𝜆𝑛+1, (43) where the function 𝑐_𝑛+1(𝑡)>0 is continuous and the constant 𝜆_𝑛+1∈(0,1).(i)If 𝜆₀>max_{1≤𝑖≤𝑛+1}{𝜆_𝑖}, then for any solution 𝑢(𝑡) of the problem (42), |𝑢(𝑡≤𝐴)|₀(𝑡)+^∞_𝑘=1^𝑛+1_{1′,2′,…,𝑘′=1}𝜀^𝑘Γ∑^𝑘_𝑖=1𝛽_𝑖′×^𝑡₀(𝑡−𝑠)^{∑𝑘𝑖=1𝛽𝑖′−1}𝐴₀(𝑠)𝑑𝑠^1/𝜆0,𝑡∈[0,𝑇], (44) where 𝐴₀(𝑡)=|𝛿|𝑡^{𝛽𝑛+1−1}Γ𝛽_𝑛+1+1Γ𝛽_𝑛+1^𝑡₀(𝑡−𝑠)^{𝛽𝑛+1−1}𝑓(+𝑠,0)𝑑𝑠_𝑛+1𝑖=11Γ𝛽_𝑖𝜆1−^𝑖𝜆₀𝜆_𝑖𝜀𝜆₀^{𝜆𝑖/(𝜆0−𝜆𝑖)}×^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)^{𝜆0/(𝜆0−𝜆𝑖)}𝑑𝑠. (45) Here 𝜀 is an arbitrary given positive number.(ii)If 𝜆₀≤min_{1≤𝑖≤𝑛+1}{𝜆_𝑖}, then the continuous solution of problem (42) is unique.

Proof. Since 1>𝛽_𝑛+1>max_{1≤𝑖≤𝑛}{𝛽_𝑖}, we obtain the following by using Lemmas 11 and 12: 𝐼^𝛽𝑛+1_𝑅𝑓(𝑡,𝑢(𝑡))=𝐼^𝛽𝑛+1_𝑅𝐷^𝛽𝑛+1_𝑅𝑢^𝜆0(𝑡+_𝑛𝑖=1)𝐼^𝛽𝑛+1_𝑅𝐷^{𝛽𝑛+1−𝛽𝑖}_𝑅𝑐_𝑖(𝑡)𝑢^𝜆𝑖(𝑡)=𝐼^𝛽𝑛+1_𝑅𝐷^𝛽𝑛+1_𝑅𝑢^𝜆0(𝑡+_𝑛𝑖=1)𝐼^𝛽𝑖_𝑅𝐼^{𝛽𝑛+1−𝛽𝑖}_𝑅𝐷^{𝛽𝑛+1−𝛽𝑖}_𝑅𝑐_𝑖(𝑡)𝑢^𝜆𝑖(𝑡)=𝑢^𝜆0(𝑡)+𝑘_𝑛+1𝑡^{𝛽𝑛+1−1}+^𝑛_𝑖=1𝐼^𝛽𝑖_𝑅𝑐_𝑖(𝑡)𝑢^𝜆𝑖(𝑡)+𝑘_𝑖𝑡^{𝛽𝑛+1−𝛽𝑖−1}=𝑢^𝜆0(𝑡)+^𝑛_𝑖=1𝐼^𝛽𝑖_𝑅𝑐_𝑖(𝑡)𝑢^𝜆𝑖(𝑡+𝑡_{𝛽𝑛+1−1})Γ𝛽_𝑛+1𝑘_𝑛+1Γ𝛽_𝑛+1+^𝑛_𝑖=1𝑘_𝑖Γ𝛽_𝑛+1−𝛽_𝑖, (46) where 𝑘_𝑖, 𝑖=1,2,…,𝑛+1, are some constants. Therefore, 0=^𝑡₀𝑓(𝑠,𝑢(𝑠))𝑑𝑠_𝑡=0=𝐼^{1−𝛽𝑛+1}_𝑅𝐼^𝛽𝑛+1_𝑅𝑓(𝑡,𝑢(𝑡))_𝑡=0=^𝑛_𝑖=1𝐼^{1+𝛽𝑖−𝛽𝑛+1}_𝑅𝑐_𝑖(𝑡)𝑢^𝜆𝑖(𝑡)+𝐼^{1−𝛽𝑛+1}_𝑅𝑢^𝜆0(𝑡)_𝑡=0+𝑘_𝑛+1Γ𝛽_𝑛+1+^𝑛_𝑖=1𝑘_𝑖Γ𝛽_𝑛+1−𝛽_𝑖=𝛿+𝑘_𝑛+1Γ𝛽_𝑛+1+^𝑛_𝑖=1𝑘_𝑖Γ𝛽_𝑛+1−𝛽_𝑖; (47) that is, 𝑘_𝑛+1Γ𝛽_𝑛+1+^𝑛_𝑖=1𝑘_𝑖Γ𝛽_𝑛+1−𝛽_𝑖=−𝛿. (48) Integrating this equality into (46) obtains 𝑢^𝜆0(𝑡)=𝐼^𝛽𝑛+1_𝑅𝑓(𝑡,𝑢(𝑡))−^𝑛_𝑖=1𝐼^𝛽𝑖_𝑅𝑐_𝑖(𝑡)𝑢^𝜆𝑖(𝑡)+𝛿𝑡^{𝛽𝑛+1−1}Γ𝛽_𝑛+1=1Γ𝛽_𝑛+1^𝑡₀(𝑡−𝑠)^{𝛽𝑛+1−1}𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝛿𝑡^{𝛽𝑛+1−1}Γ𝛽_𝑛+1−^𝑛_𝑖=11Γ𝛽_𝑖^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)𝑑𝑠. (49) Let 𝑈(𝑡)=𝑢^𝜆0(𝑡), and applying (49), given the fact that 𝑓((𝑠≤𝑓(+𝑓((𝑠(≤𝑓(𝑠,𝑢))𝑠,0)𝑠,𝑢))−𝑓𝑠,0)𝑠,0)+𝑐_𝑛+1(𝑠)|𝑢(𝑠)|^𝜆𝑛+1, (50) obtains the following: |𝑈(𝑡)|=|𝑢(𝑡)|^𝜆0≤1Γ𝛽_𝑛+1^𝑡₀(𝑡−𝑠)^{𝛽𝑛+1−1}𝑓((𝑠|𝛿|𝑡𝑠,𝑢))𝑑𝑠+^{𝛽𝑛+1−1}Γ𝛽_𝑛+1+^𝑛_𝑖=11Γ𝛽_𝑖^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)|𝑢(𝑠)|^𝜆𝑖≤1𝑑𝑠Γ𝛽_𝑛+1^𝑡₀(𝑡−𝑠)^{𝛽𝑛+1−1}𝑓(|𝛿|𝑡𝑠,0)𝑑𝑠+^{𝛽𝑛+1−1}Γ𝛽_𝑛+1+^𝑛+1_𝑖=11Γ𝛽_𝑖^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)|𝑢(𝑠)|^𝜆𝑖=1𝑑𝑠Γ𝛽_𝑛+1^𝑡₀(𝑡−𝑠)^{𝛽𝑛+1−1}𝑓(|𝛿|𝑡𝑠,0)𝑑𝑠+^{𝛽𝑛+1−1}Γ𝛽_𝑛+1+^𝑛+1_𝑖=11Γ𝛽_𝑖^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)|𝑈(𝑠)|^{𝜆𝑖/𝜆0}𝑑𝑠. (51) If 𝜆₀>max_{1≤𝑖≤𝑛+1}{𝜆_𝑖}, then, for any 𝑖∈{1,2,…,𝑛+1}, 0<𝜆_𝑖/𝜆₀<1. According to Theorem 4, for any 𝑡∈[0,𝑇], |𝑈(𝑡)|≤𝐴₀(𝑡)+^∞_𝑘=1^𝑛+1_{1′,2′,…,𝑘′=1}𝜀^𝑘Γ∑^𝑘_𝑖=1𝛽_𝑖′×^𝑡₀(𝑡−𝑠)^{∑𝑘𝑖=1𝛽𝑖′−1}𝐴₀(𝑠)𝑑𝑠, (52) where the expression of 𝐴₀(𝑡) is shown in (45). Hence, the conclusion of (i) is derived.
In proving (ii), we assume that problem (42) has two continuous solutions 𝑢 and 𝑣. Combining with the fact that 𝑐_𝑖(𝑡)∈𝐶[0,𝑇] for any 1≤𝑖≤𝑛+1 and the boundedness of the continuous function on a closed interval, there exists a finite number 𝑀 which satisfies that, for any 𝑡∈[0,𝑇], 𝑀>max|𝑢(𝑡)|,|𝑣(𝑡)|,max_{1≤𝑖≤𝑛+1}𝑐_𝑖(𝑡). (53) Cauchy’s mean value theorem provides 𝑢^𝜆𝑖(𝑡)−𝑣^𝜆𝑖(𝑡)=𝑢^𝜆0(𝑡)−𝑣^𝜆0(𝑡⋅𝜆)^𝑖⋅𝜉^𝜆𝑖_𝑖⁻¹𝜆₀⋅𝜉^𝜆0_𝑖⁻¹=𝜆_𝑖𝜆₀𝑢^𝜆0(𝑡)−𝑣^𝜆0(𝑡)⋅𝜉_𝑖^{𝜆𝑖−𝜆0}, (54) where 𝜉_𝑖, 𝑖=1,2,…,𝑛+1, are the numbers between 𝑢(𝑡) and 𝑣(𝑡). The following estimation is deduced by applying (53) and the hypothesis of 𝜆₀≤min_{1≤𝑖≤𝑛+1}{𝜆_𝑖} in (ii): 𝑢^𝜆𝑖(𝑡)−𝑣^𝜆𝑖(𝑡≤𝜆)^𝑖𝜆₀𝑢^𝜆0(𝑡)−𝑣^𝜆0(𝑡)⋅𝑀^{𝜆𝑖−𝜆0}, (55) holds for any 𝑡∈[0,𝑇] and 𝑖=1,2,…,𝑛+1. Therefore, (49), (53), and (55) obtain 𝑢^𝜆0(𝑡)−𝑣^𝜆0(𝑡=1)Γ𝛽_𝑛+1^𝑡₀(𝑡−𝑠)^{𝛽𝑛+1−1}×𝑓((𝑠((𝑠−𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_𝑛𝑖=11Γ𝛽_𝑖^𝑡₀(𝑡−𝑠)^𝛽𝑖−1𝑐_𝑖(𝑠)×𝑢^𝜆𝑖(𝑠)−𝑣^𝜆𝑖(𝑠≤)𝑑𝑠_𝑛+1𝑖=11Γ𝛽_𝑖^𝑡₀(𝑡−𝑠)^𝛽𝑖−1×𝑐_𝑖(𝑠)𝑢^𝜆𝑖(𝑠)−𝑣^𝜆𝑖(𝑠≤)𝑑𝑠_𝑛+1𝑖=1𝜆_𝑖𝑀^{1+𝜆𝑖−𝜆0}𝜆₀Γ𝛽_𝑖^𝑡₀(𝑡−𝑠)^𝛽𝑖−1×𝑢^𝜆0(𝑠)−𝑣^𝜆0(𝑠)𝑑𝑠. (56) According to Theorem 2, 𝑢^𝜆0(𝑡)−𝑣^𝜆0(𝑡)≤0, (57) which means that 𝑢(𝑡)=𝑣(𝑡),𝑡∈[0,𝑇]. (58) This completes the proof of (ii).

Moreover, we can also address the following initial value problem with the Hadamard type fractional derivative: ^𝑛_𝑖=1𝐷^{𝛼𝑛+1−𝛼𝑖}_𝐻𝑑_𝑖(𝑡)𝑢^𝛾𝑖(𝑡)+𝐷^𝛼𝑛+1_𝐻𝑢^𝛾0(𝑡)=𝑔(𝑡,𝑢(𝑡))_𝑛𝑖=1;𝐼^{1+𝛼𝑖−𝛼𝑛+1}_𝐻𝑑_𝑖(𝑡)𝑢^𝛾𝑖(𝑡)+𝐼^{1−𝛼𝑛+1}_𝐻𝑢^𝛾0(𝑡)_𝑡=1=𝜂, (59) where all functions are continuous. 𝑑_𝑖(𝑡)>0 and the constants 𝛾_𝑖,𝛼_𝑖>0 (𝑖=1,2,…,𝑛). Also 𝛾₀>0, 𝜂∈ℝ, and 1>𝛼_𝑛+1>max_{1≤𝑖≤𝑛}{𝛼_𝑖}. For any 𝛼∈(0,1) and a continuous function 𝑤, the operators 𝐷^𝛼_𝐻 and 𝐼^𝛼_𝐻 are presented below (see [18, page 110]): 𝐷^𝛼_𝐻𝑤(𝑡)=1Γ()𝑡𝑑1−𝛼𝑑𝑡^𝑡₁𝑡ln𝑠^−𝛼𝑤(𝑠)𝑠𝐼𝑑𝑠;_𝛼𝐻𝑤(𝑡)=1Γ(𝛼)^𝑡₁𝑡ln𝑠^𝛼−1𝑤(𝑠)𝑠𝑑𝑠. (60) We are able to deduce the following result by Theorems 3 and 5.

Theorem 14. Suppose that, for any 𝑡∈[1,𝑇] and 𝑦,𝑧∈ℝ, |𝑔(𝑡,𝑦)−𝑔(𝑡,𝑧)|≤𝑑_𝑛+1(𝑡)|𝑦^𝛾𝑛+1−𝑧^𝛾𝑛+1|, where the function 𝑑_𝑛+1(𝑡)>0 is continuous and the constant 𝛾_𝑛+1∈(0,1).(i)If 𝛾₀>max_{1≤𝑖≤𝑛+1}{𝛾_𝑖}, then, for any solution 𝑢(𝑡) of problem (59), |𝑢(𝑡≤)|𝐴₀(𝑡)+^∞_𝑘=1^𝑛+1_{1′,2′,…,𝑘′=1}𝜀^𝑘Γ∑^𝑘_𝑖=1𝛼_𝑖′×^𝑡₁𝑡ln𝑠^{∑𝑘𝑖=1}𝐴₀(𝑠)𝑑𝑠𝑠^1/𝛾0,𝑡∈[1,𝑇], (61) where 𝐴₀(𝑡)=𝜂(ln𝑡)^{𝛼𝑛+1−1}Γ𝛼_𝑛+1+1Γ𝛼_𝑛+1^𝑡₁𝑡ln𝑠^{𝛼𝑛+1−1}𝑔(𝑠,0)𝑑𝑠𝑠+^𝑛+1_𝑖=11Γ𝛼_𝑖𝛾1−^𝑖𝛾₀𝛾_𝑖𝜀𝛾₀^{𝛾𝑖/(𝛾0−𝛾𝑖)}×^𝑡₁𝑡ln𝑠^𝛼𝑖−1𝑑_𝑖(𝑠)^{𝛾0/(𝛾0−𝛾𝑖)}𝑑𝑠𝑠. (62) Here 𝜀 is an arbitrary given positive number.(ii)If 𝛾₀≤min_{1≤𝑖≤𝑛+1}{𝛾_𝑖}, then the continuous solution of problem (59) is unique.

The proof procedure is similar to that of Theorem 13. Thus, the procedure is omitted here.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the referees for their detailed and helpful suggestions for revising this paper. This work was supported by the National Natural Science Foundation of China (Grant nos. 11226167 and 11361020).